Greg Egan

I am a science fiction writer and computer programmer.
Latest novel: MORPHOTROPHIC.
Latest collection: SLEEP AND THE SOUL.
Web site: gregegan.net

Web site: https://www.gregegan.net
Location: Perth, Western Australia

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gregegansf_at_mathstodon.xyz@momostr.pink

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relay.damus.io

nos.lol

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Last Notes

2026-09-05 00:15:04 UTC

My favourite map of the world is ... a globe.
My family was not wealthy, but when I was a child we had, not just a world atlas with its flat maps, but a world globe made of cardboard (from Reader’s Digest, I think). So I did not grow up confused about the size of Greenland versus Africa.
So, let there be more cheap, cardboard world globes. And online, let there be more animated and interactive images of the world as it actually is: spherical. I’m as fascinated as anyone by the merits of different map projections, but most people are not 15th century navigators having to choose a projection for their flat paper charts.
https://www.abc.net.au/news/2026-09-05/africa-bigger-us-smaller-united-nations-adopts-new-world-map/107119646

2026-08-20 04:00:14 UTC

Wow, not only have astronomers found a star that comes so close to the black hole at the centre of our galaxy that a detailed study of its orbit over the next decade should be able to quantify the black hole’s spin, there is a chance that future spectroscopic analysis will show that it was once part of a binary pair with a known hypervelocity star moving away from the galaxy’s centre, providing some evidence for the capture/eject Hills mechanism, explaining how stars end up so close to a black hole in the first place.
https://www.science.org/content/article/fastest-known-star-could-reveal-spin-our-galaxy-s-giant-black-hole

2026-08-16 21:53:09 UTC

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His book-length proof looks like it’s beautifully written. Shorter than “War and Peace”, and probably more interesting.
https://arxiv.org/abs/1501.05438

2026-08-12 11:27:18 UTC

2002: I buy a prepaid SIM card, giving my name as “Greg Egan” on the account application. Asked for ID, I show my passport, which gives my first name as “Gregory” and includes my middle name.
Human checking this (who, being a human, understands how humans use their informal and full legal names): Everything is fine, enjoy your new mobile phone!
2026: Telco’s automated message to me: We must verify your identity. Please supply the details of a suitable ID document.
Me: Supplies the details on my passport.
Verification algorithm: Details do not match your account.
So I am now looking forward to hours of time wasted on getting through to a human in the telco who knows exactly the right solution to this “problem”.
And I know that people with non-Anglophone names face this kind of needless idiocy a hundred times more often.
Automating paperwork and bureaucracy is great, but 3 decades into the process, why are we still so crap at handling even the simplest, most obvious pitfalls?
[LLMs won’t fix this, they’ll just change the failure modes.]

2026-08-06 12:27:47 UTC

Not sure why I need to provide my parents’ occupations on the form for donating my body to the local medical school. It’s not as if we’re living in a medieval village where John, son of the blacksmith John, needs to be carefully distinguished from John, son of the innkeeper John.

2026-07-29 23:37:34 UTC

This is a great account of the perils of trying to reach statistical conclusions from scans of old astronomical photographic plates, when the noise and artifacts in the data turns out to be very far from uniformly distributed.
https://www.iheart.com/podcast/105-daniel-and-kellys-extraord-29862087/episode/did-aliens-put-satellites-into-earths-338322285/

2026-07-27 08:27:57 UTC

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The formal definition of the Burau representation is a bit daunting, but Vaughan Jones gives a vivid intuitive description. If you think of a braid as a bowling alley with n lanes that can cross over and under each other, and at every crossing there is a probability t of a ball in the upper lane falling down to the lower lane, the (i,j) entry in the Burau matrix is just the probability that a ball starting in lane i exits in lane j.
[This metaphor only works for braids that can be built by stringing together any sequence of the “standard generators” where strand i crosses over strand i+1. To get the whole group you need to throw in the inverses of those generators as well.]

2026-07-27 05:37:08 UTC

While the Fields Medal celebrates the under-40s, the mathematician Joan Birman has, at the age of 99, solved a major open problem in representations of the Braid groups, a topic she has worked on for more than 60 years.
An element of the Braid group 𝐵ₙ is a way of joining two sets of n points with strings, where two elements are the same if the strings can be moved around and/or stretched to follow the same paths in 3-dimensional space while keeping their endpoints fixed.
The Burau representation of 𝐵ₙ associates a matrix with each element of the group, in such a way that multiplying these matrices is the same as joining braids together to make new ones.
A representation is “faithful” if different elements of the group correspond to different matrices. The Burau representation was known to be faithful for n=1,2,3, and *not* faithful for n≥5, but the n=4 case has only now been settled:
“The Burau representation of the braid group is faithful for n = 4”
Vasudha Bharathram, Joan S. Birman, Tara E. Brendle
https://arxiv.org/abs/2607.05283
I learned this from:
https://www.scientificamerican.com/article/how-a-99-year-old-mathematician-unraveled-a-century-old-braid-mystery/
See also:
https://en.wikipedia.org/wiki/Braid_group
https://en.wikipedia.org/wiki/Burau_representation
And in case you were wondering, another group tried using chatbots on this, and failed:
“I was introduced to the problem,” He says, “by Emmanuel Breuillard and Sasha [Oleksandr] Kosyak,” two well-known mathematicians who have worked on the quandary for more than 20 years. The group spent six futile months trying to prompt various artificial intelligence chatbots using Birman’s approach. “Then, boom, on a Wednesday morning, Joan Birman herself with her two talented collaborators, claimed that they had solved the problem,” He says. “And they had.”

2026-07-25 13:23:46 UTC

My new startup, ByteShower, plans to install miniature data centres in the hot water systems of a million households by 2030. And not only will we pay all your water and electricity bills, we’ll upgrade your internet to a high-capacity fibre connection. Win-win!

2026-07-23 07:25:32 UTC

In 1909, Arthur Wieferich proved that if
𝑥ᵖ+𝑦ᵖ=𝑧ᵖ
for an odd prime 𝑝 that does not divide positive integers 𝑥, 𝑦, or 𝑧, then
𝑝² must divide 2ᵖ⁻¹−1
Fermat’s little theorem says 𝑝 itself always divides 2ᵖ⁻¹−1.
Soon afterwards, Dmytro Grave checked every prime under 1000, and conjectured that no such “Wieferich Primes” exist!
Then in 1913, Waldemar Meissner checked 1093 …
Now that we have computers this is trivial, but in 1913 it was a slog. Meissner showed that 2³⁶⁴−1 was divisible by 1093², and since 1093 – 1 = 3 × 364, and 𝑡³−1 is divisible by 𝑡−1, that’s enough.
Later, Emil Haentzschel pointed out that 2¹⁸²+1 is divisible by 1093², which does the same job because 𝑡⁶−1 is divisible by 𝑡+1. Just how annoying this was to Meissner has not been recorded.

2026-07-20 05:46:28 UTC

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Wow, that’s interesting! As a naive onlooker, I don’t really understand why people found it likely.
Is there anything you can say about how the counterexample clashes with your original intuition?

2026-07-20 04:37:11 UTC

The Jacobian conjecture is false!
The Jacobian conjecture proposed that if you have a map F:C^n→C^n with polynomials as the components, and the determinant of the matrix of partial derivatives of F is a non-zero constant, then F will have a polynomial inverse.
https://en.wikipedia.org/wiki/Jacobian_conjecture
But someone has now found a counterexample. This is easy to check, so there’s no doubt that it’s correct.
https://x.com/__alpoge__/status/2079028340955197566
𝐹(𝑥,𝑦,𝑧) =
(𝑦²(3𝑥𝑦+4)(𝑥𝑦+1)+𝑧(𝑥𝑦+1)³, 3𝑥𝑦²(3𝑥𝑦+4)+3𝑥𝑧(𝑥𝑦+1)²+𝑦,
2𝑥−𝑥³𝑧−3𝑥²𝑦)
F has a constant Jacobian determinant of -2, but it cannot have an inverse because it is not one-to-one:
F(0, 0, -1/4) = F(1, -3/2, 13/2) = F(-1, 3/2, 13/2) = (-1/4, 0, 0)

2026-07-16 22:34:58 UTC

Editor-in-Chief of the Science journals:
“Rigorous human checking of AI-generated research papers is creating bottlenecks as publishers strive to maintain the integrity of the scientific record. The challenge is requiring even more human effort, making the whole endeavor slower and more expensive.”
https://www.science.org/doi/10.1126/science.aek5570

2026-07-16 03:01:21 UTC

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If they stick securely without any hair removal that sounds like a 10.

2026-07-16 02:56:02 UTC

Anything meant to stick to human skin should be rated by its ability to cope with the existence of body hair, on a scale from:
0 for Band-Aid Aquablock: What! You put me on your *calf*? All I can do is fall to the ground.
to:
8 for Leukosilk: Ah, mildly hirsute here, no problemo!

2026-07-14 10:34:32 UTC

Nolan fans admit to selling kidneys to pay for eye surgery to allow them to perceive the Imax 1570 version of “The Odyssey” with the full acuity the director would have wanted.
https://www.theguardian.com/film/2026/jul/14/christopher-nolan-fans-are-embarking-on-epic-journeys-to-see-the-odyssey-the-way-he-wants-them-to

2026-07-04 02:51:53 UTC

TIL that the sea-sickness medication that made the D-day landing possible — given the loss of usual supplies from Europe — was derived from a native Australian plant, the corkwood tree Duboisia myoporoides, whose properties had been known for centuries to indigenous Australians.
https://www.abc.net.au/listen/programs/no-one-saw-it-coming/dday-bush-medicine/106778814
https://www.kcl.ac.uk/news/australian-indigenous-bush-medicine-was-given-to-allied-soldiers-on-d-day
https://en.wikipedia.org/wiki/Duboisia_myoporoides

2026-06-27 13:42:07 UTC

My phone acquired a handful of odd bugs yesterday (unable to use mobile data, 3-button navigation not working, quick settings panel limited to one row) after being present [switched off] in the room where I had a CAT scan.
A factory reset got everything back to normal. I suppose it could be a coincidence, but if it wasn’t, I hope all the computing hardware crucial to the scanner itself was better shielded against whatever spillage corrupted the phone's flash storage.

2026-06-18 10:42:36 UTC

I think this is well worth reading. I've been a bit frustrated by some over-confident claims that data centres in space are literally impossible due to the cooling issues alone, and other claims that those issues are fairly minor. I am nowhere near close to being familiar with all the real-world engineering nitty gritty; I understand a fair bit of thermodynamics, but that is not enough to really settle things. FWIW, the arguments here seem well grounded in physics and the technological/economic/legal issues raised strike me as plausible.
https://robtow.substack.com/p/spacex-ipo-orbital-data-centers-and

2026-06-04 04:37:38 UTC

NO, ARTIFICIAL INTELLIGENCE IS NOT CONSCIOUS
Taken to its logical conclusion, this line of thinking is absurd—and damning.
By Ted Chiang
https://www.theatlantic.com/philosophy/2026/06/no-artificial-intelligence-is-not-conscious/687378/

2026-06-02 06:04:32 UTC

“… paint damage, floor damage, damage to a kitchen doorframe, bent poles in dishwasher racks, water damage and scratches on wooden credenzas … decorative items and books moved from shelves to drawers … ”
Airbnb hosts need to add a new clause to their agreements: no robots.
https://arstechnica.com/ai/2026/06/allegedly-trashing-airbnbs-to-test-robots-puts-startup-in-legal-trouble/

2026-06-02 05:23:46 UTC

https://www.quantamagazine.org/key-chemistry-question-answered-no-quantum-computer-required-20260529/
A fascinating article in Quanta on work to find the ground state of a key molecule in nitrogen fixation via an ingenious classical computation — despite this problem often being cited as a potential benchmark for quantum computers.
We’re in the middle of a fertiliser supply shortage that is likely to cause a crisis in food security for millions of people around the planet. I wish there was more work being done on incorporating biological nitrogen fixation into more crops, whether through genetic changes to the crops themselves, or through nitrogen-fixing bacteria that can work with more plants.
Coincidentally, I recently heard a Radio Lab program on the life of Fritz (industrial nitrogen fixation) Haber, and yeow, after the great famine-busting triumph, it was dark, dark, dark.
https://radiolab.org/podcast/180132-how-do-you-solve-problem-fritz-haber

2026-05-22 07:19:48 UTC

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Just *projecting* from a high-dimensional lattice won’t give you a quasicrystal, there’s a certain kind of filtering of the points required.
https://www.gregegan.net/APPLETS/12/deBruijnNotes.html

2026-05-20 10:18:35 UTC

“I’d created 2000 free-text responses and labelled them ‘UK’. Then I copied and pasted the exact same 2000 responses but labelled these ‘US’. Finally, I combined them to create a dataset of 4000 total responses, and jumbled them up.
Despite the responses being identical for the UK and US, Copilot produced a rich, detailed summary of how US and UK respondents differed.”
https://kucharski.substack.com/p/real-signals-or-artificial-stereotypes
H/T [@sinalana.eurosky.social](https://bsky.brid.gy/r/https://bsky.app/profile/sinalana.eurosky.social )

2026-05-12 23:55:02 UTC

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Here’s a piece of a quasicrystal with icosahedral symmetry.
I describe how to get this with the 3D version of the cut-and-project method here:
https://www.gregegan.net/APPLETS/12/deBruijnNotes.html#HD
https://media.mathstodon.xyz/media_attachments/files/116/564/293/121/447/024/original/c82f0c9f8a2ac60a.mp4

2026-05-12 03:04:41 UTC

Breaking: Google AI summary of leaked Australian budget papers confirms that $2B will be allocated to phasing out all bicycle gears that rotate in the wrong direction by 2035.

2026-05-10 07:41:10 UTC

My local library shelves its fiction by genre (Crime, Historical, F&SF, ‘Relationships’), a practice I loathe, but I’m morbidly curious to learn if anyone else’s library (a) does this, but (b) has also coined some bizarre, completely nonstandard name for those few books they’ve decided can’t be squeezed into any genre at all.
My local library has chosen “Allsorts”. Gaaah.
[I suppose they decided that “Literary Fiction” sounded too snobbish, and on that I suppose I agree with them, but this weird, infantile neologism goes to the opposite extreme. Just shelve everything alphabetically by author, put colour-coded genre indicators on the spine if you must, but let people slip serendipitously out of their silos by browsing the whole damn gamut.]

2026-05-10 04:17:04 UTC

“A study that claimed OpenAI’s ChatGPT can positively impact student learning has been retracted nearly one year after publication. The journal publisher, Springer Nature, cited “discrepancies” in the analysis and a lack of confidence in the conclusions—but not before the paper racked up hundreds of citations and made the rounds on social media.”
https://arstechnica.com/ai/2026/05/influential-study-touting-chatgpt-in-education-retracted-over-red-flags/

2026-05-09 12:03:26 UTC

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In previous cases of automating mathematical activities, we’ve usually known very explicitly exactly what we were doing: computing trig tables, enumerating and checking graphs for the 4-colour map theorem, etc. This makes it relatively easy to understand the time and memory resources required, and how far they can be scaled up before hitting computational complexity limits.
It’s much less clear what LLMs are doing when arriving at these kinds of proofs, but it would be helpful if the people building and studying LLMs could make it equally transparent.
My guess is that the “algorithm” being effectively performed here (though of course not explicitly programmed as such) is something roughly along the lines of:
• extract a set P of relevant / related proofs from the literature
• break P down into a set of suitably fine-grained steps, S
• iterate through subsets of S looking for any collection of steps that will solve the problem
It’s impressive that it’s possible to automate these three things at all, even as imperfectly as LLMs do them, but if this description is more or less correct then it will eventually hit the same kind of scaling issues as any other combinatorial approach.
It might be possible to make this kind of smart-brute-force search more smart and less brutish in the manner of Alpha Go etc., but strategically pruning the decision trees for assembling valid proofs is likely to be much tougher than it was for Go.

2026-05-05 21:56:39 UTC

Breaking: Rubio, Araghchi announce negotiations to divide up parallel worlds between those in which US, Iranian conditions are met to each party’s complete satisfaction.

2026-05-05 06:04:51 UTC

I rarely used to weigh myself, but I recently started doing so daily, and I was surprised by how much the results fluctuated.
In this study:
https://pmc.ncbi.nlm.nih.gov/articles/PMC7192384/
they found:
“Within-week fluctuations of 0.35% were observed, characterised by weekend weight gain and weekday reduction which differed between all groups.”
But I seem to have peak-to-peak fluctuations closer to 4%, or ten times as much as the 0.35% seen in that study!
My food intake and exercise patterns are pretty constant, apart from doing a longish run every Sunday.
I’m curious as to what kind of fluctuations other people have measured on this timescale — especially when they’re not knowingly changing any of the relevant factors.
https://media.mathstodon.xyz/media_attachments/files/116/520/448/461/021/767/original/fcc5c12d7c6ecb20.png

2026-04-21 08:09:03 UTC

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FWIW, it seems possible that it was once spelled “cue”.
cue | kjuː |
noun
a long straight tapering wooden rod for striking the ball in snooker, billiards, etc..
origin
mid 18th century (denoting a long plait or pigtail): variant of queue.

2026-04-20 12:06:00 UTC

I’m enjoying They Might Be Giants’ new album “The World is to Dig”, though it’s a mixture of tracks I can take in after just one casual listen, and others where I need to sit down and read the lyrics from the song sheet and follow every word.
But this sweet, slightly unsettling song, with a video like a fragment of a Lynchian dream, was one of my instant favourites.
hi ho, hi ho, into the canyon we go
the ruts in the road won’t guide us back home
https://www.youtube.com/watch?v=75-1ipEs59c

2026-04-19 02:42:55 UTC

Collaboratively solving a word puzzle with a relative yesterday:
They: Ruth.
Me: That’s not a word!
They: You can be ruthless, so …
Me: That makes sense, but English is not that logical!
Concise dictionary: No “ruth”.
Complete dictionary: Archaic now, but yes, “ruth”: a feeling of pity, distress, or grief.

2026-04-09 03:18:36 UTC

Multiple lensed images of supernovae help pin down the Hubble constant:
“If SN Requiem reappears this year, it will lead to a higher value of the Hubble constant and validate the astronomers’ camp. If the supernova is not seen until next year, it will lead to a lower value…”
It’s astonishing that we can (sometimes) see the same supernova *billions* of light years away along multiple gravitationally-lensed paths that produce time differences on the human scale of 10 years or so.
https://www.science.org/content/article/how-fast-universe-expanding-cosmic-illusions-may-hold-answer

2026-04-08 21:39:55 UTC

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Not sure if you’re joking, but the book’s title is a bit of a giveaway.
https://en.wikipedia.org/wiki/Parallel_transport#Approximation:_Schild's_ladder

2026-04-08 21:34:14 UTC

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Parallel transport?

2026-03-10 12:38:48 UTC

Doctor: I’m not sure what’s going on.
Me: So, time to send Olivia Wilde and Kal Penn to break into my house to search for drugs, toxins and mould, while gossiping about their colleagues’ sex lives?
Doctor: Nah. I think we’ll do a bone marrow biopsy.
Me: It was worth a try.

2026-02-27 12:26:26 UTC

RE: https://mastodon.social/@verge/116139531198626533
“Amodei mentioned that “partial autonomous weapons … are vital to the defense of democracy” and that fully autonomous weapons may eventually “prove critical for our national defense,” but that “today, frontier AI systems are simply not reliable enough to power fully autonomous weapons.” He did not rule out Anthropic acquiescing to the military’s use of fully autonomous weapons in the future but mentioned that they were not ready now.”
What fucking deranged fantasy world are these clowns living in. None of this garbage should even be close to anything where lives are at stake.
#note1eu4…ddle

2026-02-23 11:55:15 UTC

Tomorrow I will be taking a drug to slow down my heart sufficiently for it to be imaged in a CT scan, which seems like the ideal opportunity to also cheat at snooker.

2026-01-29 04:30:34 UTC

“Add personality to your work with new editable shapes!”
— Apple, fondly imagining that this prospect is so alluring that I will pay them A$19.99/month for “Creator Studio” when the current apps I have (Pages, Numbers and Keynote) work perfectly well.

2025-12-04 13:57:15 UTC

A grim, but fascinating, unsolved mystery. A mostly peaceful and prosperous farming culture known as LBK spread across Europe from Anatolia, from 11,000 to 7,000 years ago. When it eventually came to an end, in many places the transition appears to have been unremarkable, but in some sites there are vast mass graves, and other signs of strange tensions arising in various settlements.
https://www.science.org/content/article/headless-bodies-hint-why-europe-s-first-farmers-vanished

2025-12-04 13:26:01 UTC

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At the linked post, it says:
“range of y coordinates is about +/-5537521000”

2025-11-27 07:13:54 UTC

Black Friday Sale! Our 1,000-dimensional magic golden spheres[*] are discounted by a whopping 50%!
* Discounted 1,000-dimensional magic golden spheres are 99% the radius of normally priced spheres, which may result in a small decrease in magicness and total gold volume.

2025-11-24 04:28:43 UTC

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I’m still a bit curious as to why you mentioned credit cards! I guess it can’t be a coincidence that they have exactly the same aspect ratio as a sheet of foolscap, and in the imperial measurements the specified sizes are exactly 4:1.
I’m used to all the cool relationships between the sizes of paper in the ISO 216 standard, but I never knew about the credit card / foolscap link.

2025-10-30 23:56:25 UTC

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Yes, but more of what?
The trouble with this kind of second-hand, context-free quote is that the speaker is not around to stop people twisting their words. I’m not accusing you of doing that, but honestly I hate the whole cottage industry of mining famous people’s utterances and dredging up fragments of speech or writing that they never intended to be treated as epigrams as if they are some kind of distillation of wisdom.

2025-10-27 15:03:00 UTC

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The differential equation for φ as a function of u=M/r for a null geodesic with impact parameter b M is:
dφ/du = ±1/√[1/b^2 + (2u–1)u^2]
This is derived as eqn (21) on this page:
https://www.gregegan.net/PLANCK/Tour/TourNotes.html
but it is equivalent to Misner, Thorne and Wheeler’s eqn (25.27).
For b>3√3 light can’t enter the black hole, but for any b<3√3 this describes a possible null geodesic.
I haven’t been able to find an analytic expression for Δφ as u goes from 1/2 (i.e. the event horizon) to ∞ (the singularity) for the limit as b approaches 3√3 from below, but numerical integration seems to put it at around 2.63 radians.

2025-10-19 11:57:28 UTC

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https://media.mathstodon.xyz/media_attachments/files/115/400/703/445/278/617/original/889ef75256806667.png

2025-09-15 22:28:03 UTC

There’s a new Hal Hartley movie, “Where to Land”.
https://thefilmstage.com/where-to-land-review-hal-hartley-returns-with-shaggy-and-spellbinding-drama/

2025-09-15 08:52:02 UTC

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TextEdit, XCode, Mail, Pages and Safari all update their search string after it’s changed in another app.
Like you, I don’t see any sharing of the search string in/out of Finder or Music.

2025-09-15 06:55:03 UTC

In MacOS, most apps share a common search string: if you search for “lion” in app 1 then search for “tiger” in app 2, then return to app 1, the app 1 search box will now contain “tiger”.
Is this behaviour:

2025-09-13 14:03:22 UTC

If you want to feel good about humanity for 2½ hours, “Thirteen Lives” is a tense, moving, inspiring tale of the 2018 Thai cave rescue, when ingenuity, bravery, cooperation, desperation and luck paid off. The bitter aftertaste is how many children we’ve collectively failed since.

2025-09-04 05:53:52 UTC

The Bertrand “paradox” is the famous observation that asking for the probability that a “random chord” on a circle spans more than 120° is meaningless until you specify exactly how you choose the chord. The Wikipedia article describes three methods that give P = 1/3, 1/2 and 1/4.
So … having played around with a lot of calculations for what happens when you pick two points inside a set and draw a line through them, I couldn’t resist asking what happens if you choose a “random chord” that way.
The answer is:
P(θ > 120°) = 1/3 + (3√3)/(4π) ≈ 0.74683
https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

2025-09-02 14:00:40 UTC

A triangle has edges of length 2, 3 and 4.
Pick two points P and Q at random in the interior of the triangle, and draw the line L that contains both P and Q.
Which two edges of the triangle is the line L most likely to intersect?

2025-09-02 06:33:15 UTC

Zuckerberg: I am *literally* building Roko’s Basilisk! Everyone who assists me will become a billionaire, and everyone who chooses not to will be tortured for eternity!
Everyone: Nah, we’re good. Also, no sane adult takes that Yoko’s Broccoli thing seriously.

2025-09-02 03:14:32 UTC

TIL that if you were a premature baby in New York in the 1900s, your best chance for first class neonatal care would come from being exhibited in a state of the art incubator as an attraction at Coney Island.
https://www.abc.net.au/listen/programs/no-one-saw-it-coming/no-one-saw-it-coming-coney-island-babies-fake-doctor/105669698

2025-08-28 08:20:49 UTC

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I wanted to celebrate by posting animations of the cube and octahedron being Rupert, but Mastodon keeps rejecting the GIFs, though they’re only about 1Mb.
So if anyone wants to see them, here are the links:
https://www.gregegan.net/images/RupertCube.gif
https://www.gregegan.net/images/RupertOctahedron.gif

2025-08-26 03:28:14 UTC

One formula to rule them all!
P(d, F) = 4d / ((d+1) F)
In d-dimensional space, d≥2, given a regular polytope with F faces that come in opposite pairs, if you pick two points at random from the interior of the polytope and draw the line that contains them, the probability that the line will intersect both faces of any of the F/2 opposite pairs is P(d, F).
For d=2, this applies to all polygons with an even number of edges.
For d=3, the four Platonic solids apart from the tetrahedron.
For d=4, the hypercube, 16-cell, 24-cell, 120-cell and 600-cell.
For d>4, the hypercubes and the cross-polytopes.

2025-08-25 12:56:21 UTC

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“For even F” should be “For F greater than 4, i.e. excluding the tetrahedron, which doesn’t have pairs of opposite faces.”
I did have to do it case by case! When I get a chance I’ll look more closely at the calculations and see if I can isolate the underlying reason.

2025-08-25 08:51:42 UTC

∑ᵢ₌₀ⁿ⁻²∑ⱼ₌ᵢ₊₁ⁿ⁻¹𝑠𝑖𝑛((𝑗−𝑖)π/𝑛)⁴=3𝑛²/16
Well, obviously.

2025-08-24 14:07:47 UTC

@npub1rm9…r6dx In a rectangle the opposite vs adjacent probabilities are the same as for a square.

2025-08-24 12:30:06 UTC

Pick two points uniformly at random inside a regular n-gon.
For even n≥4, the probability that the line containing them will pass through two opposite sides of the n-gon is:
8/(3n)
For odd or even n≥3, the probability that the line will pass through two adjacent sides is:
16 sin(π/n)^4 /(3n)

2025-08-24 08:32:20 UTC

Robert Musil, “A Man Without Qualities”
https://media.mathstodon.xyz/media_attachments/files/115/082/806/326/718/346/original/26bc0ba7909f3ed3.jpg

2025-08-23 22:01:22 UTC

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The solution for d=2 is given here:
https://mathstodon.xyz/@gregeganSF/115076414261569820
The general solution can be found the same way, by noting that the conditions on x₁ and x₂ for fixed values of y₁ and y₂ apply independently to all of the d-1 other coordinates. So the probability we used for d=2, |y₁-y₂|, just becomes |y₁-y₂|^{d-1}.
We then integrate |y₁-y₂|^{d-1} over the square 0 ≤ y₁, y₂ ≤ 1, or double the integral over the triangle where y₁ ≥ y₂. This can be done with a change of variables to:
m = y₁-y₂
p = y₁+y₂
y₁ = (p+m)/2
y₂ = (p-m)/2
The change of variables gives a factor of 1/2, and we integrate:
(1/2) m^{d-1} for p from p=m (i.e. y₂=0) to p=2-m (i.e. y₁=1).
This gives us (1-m) m^{d-1}, which we then integrate for m from 0 to 1, to get:
1/[d(d+1)]
We double to get the integral over the full square, then multiply by d to account for all pairs of opposite sides. So:
P(d) = 2/(d+1)
and:
P(17) = 1/9

2025-08-23 11:27:39 UTC

Here’s some Gemini gaslighting:
• giving a wrong answer to a puzzle
• giving Python code that could test its claim
• asserting it obtained results from the code supporting its claim
• but when I ran the code myself, it showed the claim was false.
https://g.co/gemini/share/2188cebf58cc

2025-08-23 10:32:09 UTC

You wake up in a universe with a different number of spatial dimensions.
Your senses are confused, and you’re too embarrassed to ask someone on the street “Hey, how many dimensions does space have?”
But then you chance upon someone who likes mathematical puzzles, and ask them, “If I picked two points at random inside a hypercubic crate, what would the probability be that the line joining them intersected two opposite sides of the crate?”
They scribble on some paper for a while, then reply “One ninth.”
How many dimensions does space have here?

2025-08-22 13:12:02 UTC

Here’s a fun puzzle I heard from @npub1k8l…6pkl
Pick two points uniformly at random in a square. What is the probability that the line that contains both points intersects two opposite edges of the square, rather than two adjacent edges?

2025-08-22 07:37:21 UTC

A letter from my local council announcing plans for a “Montorsoli school” made me wonder if it was a typo for “Montessori” … but on closer scrutiny, a religious order really is planning to name this facility after the 16th century Florentine sculptor & friar Giovanni Montorsoli.

2025-08-16 14:32:12 UTC

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The character is just seen wearing it; there is no plot related to it. It makes no sense that he would go out of his way to have it made, as he has absolutely no interest in physics. Maybe he could have stumbled on it in a second-hand store and liked the design.
I just wondered if there was a real-world trend of Californian physics grad students wearing Peskin and Schroeder T-shirts, which might at least have slightly raised the probability that a Hollywood costume designer on a comedy show that has nothing to do with physics could have seen one in the wild, and liked it enough to decide to dress a character with no interest in physics in such a T-shirt.

2025-08-16 13:33:59 UTC

I read some reviews praising “Platonic”, and tried watching a few episodes, but it’s not my thing. But I got as far as episode 5, where Seth Rogen’s craft-beer-blah character who knows as much about physics as I do about craft beer is inexplicably wearing a T-shirt of the cover of Peskin and Schroeder’s “An Introduction to Quantum Field Theory”.
Is that really a thing, or was the costume designer just playing a strange joke? No such T-shirt showed up when I googled. Do people who actually study QFT wear Peskin and Schroeder T-shirts?

2025-08-16 11:23:52 UTC

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I love the fact that 106 years ago Eddington struggled to detect a minuscule amount of gravitational lensing by the sun during an eclipse, but these days every tenth Hubble image of a massive galaxy needs a footnote saying which parts of the picture are the actually galaxy and which are the hugely magnified image of another, far more distant galaxy that just happens to lie behind it.
When I travel back in time to visit Einstein on his death bed and console him about his unified theory not working out, I hope to cheer him up with a collection of Einstein rings, along with a compendium of gravitational wave recordings.

2025-08-09 16:03:47 UTC

Me: Ha, take that, complicated blue curve! I have enclosed you in a circle, and now all my calculations putting bounds on your behaviour will be vastly simpler!
Complicated blue curve: Have you actually proved that the circle encloses me?
Me: I hate you very much.
https://media.mathstodon.xyz/media_attachments/files/114/999/638/506/644/354/original/8ee26ed932f7d7a6.png

2025-07-29 11:19:31 UTC

You might have heard that the elliptical orbits of a planet around a star under Newtonian gravity have a kind of *four*-dimensional symmetry. There is a lot of beautiful mathematics behind the ideas that make this notion precise, but here is a simple way to get a handle on the underlying symmetry.
Choose two vectors A and B whose lengths are equal. Place one focus of an ellipse at the origin, place the second focus at A + B, and position the two points of the ellipse where the tangent to the curve is parallel to its axis at A and B. This construction will always yield an ellipse with a semi-major axis equal to |A| (with the degenerate case of a line when A = B), and the entire family of ellipses with one focus at the origin and the same length for the semi-major axis can be generated by rotating A and B independently.
The total energy of a Newtonian orbit like this depends *only* on the semi-major axis of the ellipse, so what this tells us is that there are *two* three-dimensional rotations that we can perform while still yielding an orbit with the same energy. And the set of all rotations of *four*-dimensional space correspond to *pairs* of three-dimensional rotations.
For much, much more on all this:
https://www.gregegan.net/SCIENCE/Ellipse/Ellipse.html
and a beautiful series of blog posts by @npub1nf4…nqe4
https://johncarlosbaez.wordpress.com/2025/07/14/the-kepler-problem-part-2/
https://media.mathstodon.xyz/media_attachments/files/114/936/227/935/212/260/original/b336f7f13c81f68f.png

2025-07-18 05:16:33 UTC

“[The Arctic tern] sees two summers per year and more daylight than any other creature on the planet.”
“A tern ringed as an unfledged chick on the Farne Islands in the northern summer of 1982 reached Melbourne just three months after fledging – a journey of more than 22,000 km.”
https://en.wikipedia.org/wiki/Arctic_tern

2025-07-13 08:30:42 UTC

“In the past two years, however, with surprisingly little notice, renewable energy has suddenly become the obvious, mainstream, cost-efficient choice around the world.”
https://www.newyorker.com/news/annals-of-a-warming-planet/46-billion-years-on-the-sun-is-having-a-moment

2025-07-07 22:34:34 UTC

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I’d be amazed if there is a way to do better than that, though.

2025-06-22 06:44:40 UTC

I recently bought a stainless steel pressure cooker [to replace an old aluminium one that can’t be used with an induction cooktop], so for the first time in my life I now own a cylindrical object with a highly polished mirrored surface.
When I went looking online for examples of cylindrical mirror anamorphic art (i.e. images that appear distorted until you view them reflected in a cylinder), I discovered a sculptor, Jonty Hurwitz, who does this for solid objects rather than just flat images.
https://www.thisiscolossal.com/2013/01/the-skewed-anamorphic-sculptures-and-engineered-illusions-of-jonty-hurwitz/
[The antihero in my story “Crisis Actors” uses cylindrical mirror anamorphism as an “encryption” technique. Wikipedia notes that “A secret mirror anamorphosis portrait of Bonnie Prince Charlie, held at the West Highland Museum, can only be recognized when a polished cylinder is placed in the correct position. To possess such an image would have been seen as treason in the aftermath of the 1746 Battle of Culloden”.]
https://media.mathstodon.xyz/media_attachments/files/114/725/633/427/623/829/original/355c41174eb5d5b8.jpg

2025-05-25 02:37:04 UTC

Avoiding a crush in a crowd is more likely if people are encouraged to adopt two strategies:
• Follow the direction that the minority of people in the room are choosing.
• Change your decision to another exit as soon as you find a better one, that is, one with a smaller queue.
ABC podcast: https://www.abc.net.au/listen/programs/allinthemind/do-people-behave-differently-crowd-pychology/105281052
Research paper (open access):
https://www.sciencedirect.com/science/article/pii/S0968090X24002845#s0020

2025-05-13 07:30:24 UTC

They don’t write medical articles like this anymore! But this is not Arthur Conan Doyle, it’s Henry Bence Jones.
‘On the 1st of November 1845 I received from Dr. Watson the following note, with a test tube containing a thick, yellow, semi-solid substance:—“The tube contains urine of very high specific gravity; when boiled it becomes highly opake; on the addition of nitric acid it effervesces, assumes a reddish hue, becomes quite clear, but, as it cools, assumes the consistence and appearance which you see: heat reliquifies it. What is it ?” A few hours afterwards a specimen of the same urine, passed by a grocer forty-seven years of age, who had been out of health for thirteen months, was sent to me by Dr. MacIntyre. He being in attendance on the case with Dr. Watson, had two days previously first observed the peculiar reactions of the urine.’
https://royalsocietypublishing.org/doi/10.1098/rstl.1848.0003

2025-05-07 05:38:34 UTC

My 1995 novel DISTRESS — about Theories of Everything, Anthrocosmologists, and artificial islands — was published in French in 1997.
Now le Bélial’ have brought it back into print in a revised edition!
Translated by Bernard Sigaud, cover by Aurélien Police.
https://belial.fr/legacy/a/greg-egan/l-enigme-de-l-univers_belial
English-language editions:
https://www.gregegan.net/BIBLIOGRAPHY/Bibliography.html#p53
https://media.mathstodon.xyz/media_attachments/files/114/464/931/857/325/253/original/ff54716d06497ba0.jpg

2025-05-05 13:30:45 UTC

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Yes, there are arguments to be made for two different replies, depending on precisely how you define tidal stretching. So I’m not claiming that there’s one correct answer to this puzzle, I’m just hoping people will think carefully about the whole situation.

2025-05-05 11:27:07 UTC

Here’s a fun puzzle about tidal forces.
Suppose a planet in a tidally-locked circular orbit of radius R around its star experiences tidal stretching T (measured in your favourite units for acceleration per distance).
If a wandering planet fell into this solar system on a head-on collision course for the star, what tidal stretching would it experience when its distance from the star was also R?

2025-05-05 10:49:33 UTC

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I wonder if he’d find the claim more plausible if it was fleshed out with more details, and maybe some analogies / demonstrations.
The real key to tides is not that the Moon exerts a force, per se, but that the force is greater on closer objects. A magnet does the same, though; it shouldn't be hard to convince someone of that by direct experience.
And for added fun, maybe there’s some way to include demonstrations of centrifugal force, which can only balance the gravitational attraction of the Moon on the Earth at one distance. So closer parts of the ocean experience more attraction than centrifugal force, while further parts experience more centrifugal force than gravitational attraction ... stretching the ocean into an egg shape pointing towards the Moon that the Earth rotates within.

2025-05-04 14:14:55 UTC

In base 10, there’s an integer whose reciprocal has a repeating block with all 10 digits exactly once:
1/72,728 = 0.000(0137498625) ...
I found examples for all even bases from 2 to 34 except 8, 16 and 32. And for a few days I thought “If these are the only examples, surely it can’t be hard to prove there are no others.”
But then I stepped back and looked at the simpler things in number theory people have been trying to prove for centuries.
For example: are 3, 5, 17, 257 and 65537 the only primes of the form 2^(2^n)+1?
https://en.wikipedia.org/wiki/Fermat_number
If Euler couldn’t prove this, and nobody since has managed to do it either, it seems very unlikely that I’ll ever prove a conjecture about the bases in which the reciprocal of an integer has a non-redundant pandigital reptend.
But check out base 34!
https://gregegan.net/SCIENCE/Reptends/Reptends.html
https://media.mathstodon.xyz/media_attachments/files/114/449/976/483/147/941/original/b6d0d08b7ea93622.png

2025-05-02 07:45:57 UTC

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I must confess I’m having some trouble figuring out precisely what the puzzle is asking.
I guess it is saying that there are automorphisms of the linked-keyring system that map any pair of rings to any other pair, but this is not true of triples ... but are these automorphisms purely topological bijections on the complement of the rings in R^3, or is there any geometric restriction as well? The bit about not telling left from right suggests that reflections would be valid maps, but I’m a bit nervous about the rings having a specific size and a rigid shape, as if that makes some kind of difference from the case where they were just rubber bands. Or maybe that’s a red herring.

2025-05-02 04:16:11 UTC

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They might need to escalate the prize beyond Pizza Hut gift cards.

2025-05-01 07:00:11 UTC

[Serious documentary voice]: “Observers will be paying close attention to the colour of the smoke emerging from the Sistine Chapel next week, as the conclave follows a time-honoured tradition: blue smoke means the new Pope will be a boy, while pink smoke indicates a girl.”

2025-03-29 11:05:15 UTC

Asimov's SF and Analog have put the top few stories in each category in their 2024 annual readers’ poll free to read online on their web sites.
My novella “Death and the Gorgon” from Asimov’s and my novelette “Vouch For Me” from Analog are among them.
Links:
https://www.asimovs.com/about-asimovs/readers-award-finalists/
https://www.analogsf.com/about-analog/analytical-laboratory-finalists/
If you like some of these stories, I hope you’ll consider subscribing to the magazines. It’s a tough time for all SF magazines, and they’re the place where new writers get their start.

2025-03-27 11:17:34 UTC

Curious to see what ads Netflix shows me now that they know I was born in 1900.

2025-03-26 22:18:00 UTC

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Yes, I phrased that badly; I knew the other two conditions followed from the first, so I should have said that explicitly, rather than making it sound like they were independent requirements.

2025-03-25 08:10:40 UTC

“… by focusing on tasks that are relatively easy for humans, yet hard, or impossible, for AI, we shine a spotlight on capability gaps that do not spontaneously emerge from "scaling up".”
https://arcprize.org/blog/announcing-arc-agi-2-and-arc-prize-2025

2025-03-20 08:46:56 UTC

This makes me feel very, very old, though maybe in a good way: I lived long enough to see the universe change from “Hmm, maybe heading for a Big Crunch” to “More likely everything receding into the distance behind cosmic horizons” to “WTF, honestly, who knows now?”
https://www.science.org/content/article/mystery-force-behind-universe-s-accelerating-expansion-may-not-be-so-constant-after-all

2025-03-20 07:42:50 UTC

The market for burner phones in the carparks of airports worldwide must be skyrocketing.
https://www.theguardian.com/us-news/2025/mar/19/trump-musk-french-scientist-detained

2025-03-18 13:48:11 UTC

INTERIOR. OVAL OFFICE. January 2033.
Camera tracks across the floor, then ascends to show the surface of the Resolute Desk, where two robotic autopens are fighting, signing various documents then crossing out each other’s signature, while emitting noises like angry steam irons.

2025-03-18 00:04:36 UTC

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I wasn’t arguing against anything you’re saying, I was just trying to spell out all the requirements on D(a) for the sake of performing concrete calculations.
@npub1nf4…nqe4 @npub1ums…7ra6

2025-03-17 11:26:26 UTC

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I should just add that, while the equation:
D(a c) = a D(c) + D(a)
makes the map:
x ↦ a x + b
↓
x ↦ a x + (b q + D(a))
into an automorphism (where q is any invertible element), when you *also* need it to be class-preserving and outer, if you compare this with what happens to
x ↦ a x + b
conjugated by:
x ↦ c x + d
Namely:
x ↦ a x + (b c + d (1-a))
you have the additional constraint that:
D(a) = δ(a) (1-a)
where δ(a) is not a constant function.

2025-03-16 22:50:02 UTC

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The crucial properties that D(a) needs to satisfy in order to get an automorphism are:
D(a c) = a D(c) + D(a) = c D(a) + D(c)
D(1) = 0
while D(a) = 0 for all a is ruled out, because then the automorphism is inner.

2025-03-16 05:20:06 UTC

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I think I’ve found a class-preserving outer automorphism of Aff(ℤ/8)!
You map:
x ↦ a x + b
to:
x ↦ a x + (b + D(a))
where
D(1) = D(3) = 0
D(5) = D(7) = 4
I don’t think this is unique. But I’m still searching through the (4!)^8 choices of permutations of the odd and even parts of ℤ/8, for each value of a.
Conjugation preserves the parity of b, so rather than there being 8! choices of permutations of ℤ/8 for each value of a, there are (4!)^2.

2025-03-15 23:03:42 UTC

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I checked all 4^4 possibilities, but I can’t find any functions where this is true other than the constant functions.
So either my calculations are flawed, or something trickier is happening to b that can't be achieved with multiplication.
Checking all (8!)^4 choices of four independent permutations of ℤ/8 isn’t practical ... and though I *think* the a=1 case must correspond to multiplication by 3, 5 or 7, even 3 * (8!)^3 is too big.

2025-03-15 12:03:28 UTC

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I tried to solve the puzzle, but I must be confused about something.
If you conjugate (a,b) with (c,d) I think you get (a, b c + d - a d), but most importantly the first element, a, is unchanged. So any automorphism of the kind the puzzle is describing must leave the a in x ↦ a x + b unchanged.
Suppose we try to find such an automorphism by brute force, and start by looking at what it does to the subgroup consisting of elements of the form x ↦ x + b (which it must preserve). All it can do is change b by permuting the elements of ℤ/8.
When I checked all 8! permutations of ℤ/8, the only nontrivial ones for which this is an automorphism are those that correspond to multiplication of b by 3, 5 or 7. But these are inner automorphisms, corresponding to conjugation by (3,0), (5,0) or (7,0).
I must be doing something wrong! Can you give me a hint as to where I’ve messed up?